Geometric Langlands Correspondence: Further Directions

朗兰兹纲领是一个数学框架，它统一了许多不同数学领域的问题，特别是数论和表示理论，在数论、表示理论、几何和数学物理之间有着深刻的联系，但目前还只是部分理解。经典朗兰兹程序已经被研究了50多年，并在求解经典丢芬图方程（例如费马大定理的解）方面找到了重要的应用。几何朗兰兹程序相对较新，但由于它与几何和物理等其他学科有联系，因此发展迅速，人们可以从中获得直觉。在几何朗兰兹规划中，经典朗兰兹规划中的数域被曲线及其函数域所取代，数论中的连接网被简洁表述的等价或对应所取代。该项目将扩展和发展几何朗兰对应在不同的设置，并为研究生提供研究训练的机会。更详细地说，一个项目将建立基本局部等价，这是仿射格拉斯曼群的惠特克范畴与朗兰兹对偶的Kazhdan-Lusztig范畴之间的等价，作为分解范畴。这个等价可以看作是局部几何朗兰兹规划的起点。该项目将通过将两边等同于一个组合对象来建立所需的等效性，该组合对象可以直接用根数据和量子参数表示（所谓的分解代数Omega）。另一个项目将通过实现自同构函数的空间作为Frobenius在具有幂零奇异支持的自同构轴的范畴上的范畴迹，将几何和经典朗兰兹理论（在函数场的情况下）直接联系起来。在这样做的过程中，人们很自然地通过shtukas在朗兰兹参数的过程模空间上重新推导出V. Lafforgue对自同构函数空间的分解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Langlands Program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and representation theory, in a web of deep and as yet only partially understood connections between number theory, representation theory, geometry, and mathematical physics. The classical Langlands program has been studied for more than fifty years, and has found significant applications to solving classical Diophantine equations, such as the solution of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development as it is also connected with other subjects such as geometry and physics, from where one can draw intuition. In the geometric Langlands program, the number fields in the classical Langlands program are replaced by curves and their function fields, and the web of connections in number theory is replaced by concisely formulated equivalences, or correspondences. This project will extend and develop geometric Langlands correspondences in different settings and provide research training opportunities for graduate students.In more detail, one project will establish the Fundamental Local Equivalence, which is an equivalence between the Whittaker category of the affine Grassmannian for a reductive group and the Kazhdan-Lusztig category for its Langlands dual, as factorization categories. This equivalence could be considered as the starting point for the local geometric Langlands program. The project will establish the required equivalence by equating both sides to a combinatorial object that is directly expressible in terms of the root data and the quantum parameter (the so-called factorization algebra Omega). Another project will directly relate the geometric and classical Langlands theories (in the case of function fields) by realizing the space of automorphic functions as the categorical trace of the Frobenius on the category of automorphic sheaves with nilpotent singular support. In the process of doing so, one naturally re-derives V. Lafforgue's decomposition of the space of automorphic functions over the course moduli space of Langlands parameters via shtukas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.